High Order Compact Scheme For Discontinuous Differential Equations

A high order implicit second derivative compact method is given which is similar the Adams-Moulton method, but requiring only two steps for sixth order. This method is used in both predictor-corrector and Newton's method formulations, and although the compact scheme is not A-stable or stiffly stable, it's region of stability is over six times greater than the Sixth order Adams-Moulton method. This compact method has a small truncation error coefficient, and is more accurate than Enright's method within the region of stability. Fourth and sixth order explicit compact methods are derived as well, and the three step sixth order explicit method is used as the predictor for the harmonic oscillator equation with Coulomb damping. Consistency and rate of convergence conditions are derived for these compact methods, and convergence is proved as well. There region of stability is plotted for the sixth order implicit case. The two step sixth order implicit compact method is compared against the five step stiffly-stable Enright method and the five step Adams-Moulton scheme on three test problems, and is shown to be more accurate than Enright's method, and has better accuracy and is more stable than Adams-Moulton. The predictor-corrector compact formulation is tested on two Coulomb friction problems, and the difficulties caused by the discontinuous right-hand side is avoided by breaking the problem into segments between the discontinuities.